Algorithms Array Loops – Add Unique Items to an Array Without Many Comparisons

Appendix A on p46 of NIST SP 800-63 talks about the work of Claude Shannon, who estimates password entropy using a number of bits. Indeed, this is the document that the XKCD cartoon uses to calculate the entropy bits. Specifically:

• the entropy of the first character is taken to be 4 bits;
• the entropy of the next 7 characters are 2 bits per character; this is roughly consistent with Shannon’s estimate that “when statistical effects extending over not more than 8 letters are considered the entropy is roughly 2.3 bits per character;”
• for the 9th through the 20th character the entropy is taken to be 1.5 bits per character;
• for characters 21 and above the entropy is taken to be 1 bit per character;
• A “bonus” of 6 bits of entropy is assigned for a composition rule that requires both upper case and non-alphabetic characters. This forces the use of these characters, but in many cases thee characters will occur only at the beginning or the end of the password, and it reduces the total search space somewhat, so the benefit is probably modest and nearly independent of the length of the password;
• A bonus of up to 6 bits of entropy is added for an extensive dictionary check. If the attacker knows the dictionary, he can avoid testing those passwords, and will in any event, be able to guess much of the dictionary, which will, however, be the most likely selected passwords in the absence of a dictionary rule. The assumption is that most of the guessing entropy benefits for a dictionary test accrue to relatively short passwords, because any long password that can be remembered must necessarily be a “pass-phrase” composed of dictionary words, so the bonus declines to zero at 20 characters.

The idea is that an authentication system would pick certain entropy levels as thresholds. For example, 10 bits may be weak, 20 medium and 30 strong (numbers picked arbitrarily as an example, not a recommendation). Unfortunately, the document does not recommend such thresholds, probably because the computational power available to brute force or guess passwords increases over time:

As an alternative to imposing some arbitrary specific set of rules, an authentication system might grade user passwords, using the rules stated above, and accept any that meet some minimum entropy standard. For example, suppose passwords with at least 24-bits of entropy were required. We can calculate the entropy estimate of “IamtheCapitanofthePina4” by observing that the string has 23 characters and would satisfy a composition rule requiring upper case and non-alphabetic characters.

This may or may not be what you are looking for but is not a bad reference point, if nothing else.

[Edit: Added the following.]

The paper Testing Metrics for Password Creation Policies by Attacking Large Sets of Revealed Passwords (by Matt Weir, Sudhir Aggarwal, Michael Collins and Henry Stern) demonstrated the Shannon model, described above, is not an accurate model of entropy for human generated passwords. I would recommend looking at "Section 5 Generating New Password Creation Policies" for more accurate proposals.

array.insert is O(n) for insertions in the middle (because it has to copy a part of the array), making the second variant O(n^2), while push is O(1), and sort! is O(n log n), so the first variant is only O(n log n). Also the first variant looks cleaner.